Properties

Label 504.4
Level 504
Weight 4
Dimension 8758
Nonzero newspaces 30
Sturm bound 55296
Trace bound 25

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Defining parameters

Level: \( N \) = \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 30 \)
Sturm bound: \(55296\)
Trace bound: \(25\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(504))\).

Total New Old
Modular forms 21312 8938 12374
Cusp forms 20160 8758 11402
Eisenstein series 1152 180 972

Trace form

\( 8758q - 10q^{2} - 18q^{3} + 10q^{4} + 44q^{5} + 16q^{6} + 24q^{7} - 88q^{8} - 90q^{9} + O(q^{10}) \) \( 8758q - 10q^{2} - 18q^{3} + 10q^{4} + 44q^{5} + 16q^{6} + 24q^{7} - 88q^{8} - 90q^{9} - 214q^{10} - 190q^{11} - 236q^{12} - 50q^{13} - 164q^{14} - 96q^{15} - 286q^{16} + 254q^{17} + 216q^{18} + 486q^{19} + 1412q^{20} - 228q^{21} + 1298q^{22} - 636q^{23} + 504q^{24} - 432q^{25} - 832q^{26} - 600q^{27} - 1198q^{28} + 384q^{29} - 260q^{30} + 928q^{31} - 560q^{32} + 1634q^{33} - 2242q^{34} + 1002q^{35} + 1484q^{36} - 42q^{37} + 688q^{38} - 732q^{39} - 888q^{40} - 1594q^{41} + 94q^{42} + 2786q^{43} + 210q^{44} - 2284q^{45} + 4288q^{46} + 2652q^{47} + 1568q^{48} - 1462q^{49} + 7250q^{50} + 1050q^{51} + 5428q^{52} - 1486q^{53} - 1368q^{54} - 4608q^{55} - 332q^{56} - 3170q^{57} - 7508q^{58} - 11272q^{59} - 1944q^{60} + 700q^{61} - 7916q^{62} - 1208q^{63} - 4622q^{64} + 1768q^{65} - 7616q^{66} + 2490q^{67} - 13458q^{68} + 2848q^{69} - 11592q^{70} + 7168q^{71} - 8400q^{72} + 3594q^{73} + 1214q^{74} + 12922q^{75} + 8298q^{76} + 5922q^{77} + 964q^{78} - 176q^{79} + 7944q^{80} - 1150q^{81} - 366q^{82} - 4046q^{83} + 17752q^{84} + 2792q^{85} + 24638q^{86} - 6456q^{87} + 4730q^{88} - 6106q^{89} + 5584q^{90} + 4038q^{91} + 3030q^{92} - 7996q^{93} - 2766q^{94} - 1552q^{95} - 4224q^{96} + 5814q^{97} - 17134q^{98} + 3564q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(504))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
504.4.a \(\chi_{504}(1, \cdot)\) 504.4.a.a 1 1
504.4.a.b 1
504.4.a.c 1
504.4.a.d 1
504.4.a.e 1
504.4.a.f 1
504.4.a.g 1
504.4.a.h 1
504.4.a.i 2
504.4.a.j 2
504.4.a.k 2
504.4.a.l 2
504.4.a.m 2
504.4.a.n 2
504.4.a.o 2
504.4.b \(\chi_{504}(55, \cdot)\) None 0 1
504.4.c \(\chi_{504}(253, \cdot)\) 504.4.c.a 8 1
504.4.c.b 10
504.4.c.c 16
504.4.c.d 16
504.4.c.e 20
504.4.c.f 20
504.4.h \(\chi_{504}(71, \cdot)\) None 0 1
504.4.i \(\chi_{504}(125, \cdot)\) 504.4.i.a 8 1
504.4.i.b 88
504.4.j \(\chi_{504}(323, \cdot)\) 504.4.j.a 72 1
504.4.k \(\chi_{504}(377, \cdot)\) 504.4.k.a 24 1
504.4.p \(\chi_{504}(307, \cdot)\) n/a 118 1
504.4.q \(\chi_{504}(25, \cdot)\) n/a 144 2
504.4.r \(\chi_{504}(169, \cdot)\) n/a 108 2
504.4.s \(\chi_{504}(289, \cdot)\) 504.4.s.a 2 2
504.4.s.b 2
504.4.s.c 2
504.4.s.d 2
504.4.s.e 2
504.4.s.f 4
504.4.s.g 6
504.4.s.h 6
504.4.s.i 6
504.4.s.j 8
504.4.s.k 10
504.4.s.l 10
504.4.t \(\chi_{504}(193, \cdot)\) n/a 144 2
504.4.w \(\chi_{504}(205, \cdot)\) n/a 568 2
504.4.x \(\chi_{504}(31, \cdot)\) None 0 2
504.4.y \(\chi_{504}(173, \cdot)\) n/a 568 2
504.4.z \(\chi_{504}(95, \cdot)\) None 0 2
504.4.be \(\chi_{504}(139, \cdot)\) n/a 568 2
504.4.bf \(\chi_{504}(115, \cdot)\) n/a 568 2
504.4.bk \(\chi_{504}(19, \cdot)\) n/a 236 2
504.4.bl \(\chi_{504}(17, \cdot)\) 504.4.bl.a 48 2
504.4.bm \(\chi_{504}(107, \cdot)\) n/a 192 2
504.4.br \(\chi_{504}(155, \cdot)\) n/a 432 2
504.4.bs \(\chi_{504}(257, \cdot)\) n/a 144 2
504.4.bt \(\chi_{504}(11, \cdot)\) n/a 568 2
504.4.bu \(\chi_{504}(41, \cdot)\) n/a 144 2
504.4.bz \(\chi_{504}(239, \cdot)\) None 0 2
504.4.ca \(\chi_{504}(5, \cdot)\) n/a 568 2
504.4.cb \(\chi_{504}(23, \cdot)\) None 0 2
504.4.cc \(\chi_{504}(293, \cdot)\) n/a 568 2
504.4.ch \(\chi_{504}(269, \cdot)\) n/a 192 2
504.4.ci \(\chi_{504}(359, \cdot)\) None 0 2
504.4.cj \(\chi_{504}(37, \cdot)\) n/a 236 2
504.4.ck \(\chi_{504}(199, \cdot)\) None 0 2
504.4.cp \(\chi_{504}(223, \cdot)\) None 0 2
504.4.cq \(\chi_{504}(277, \cdot)\) n/a 568 2
504.4.cr \(\chi_{504}(103, \cdot)\) None 0 2
504.4.cs \(\chi_{504}(85, \cdot)\) n/a 432 2
504.4.cx \(\chi_{504}(185, \cdot)\) n/a 144 2
504.4.cy \(\chi_{504}(347, \cdot)\) n/a 568 2
504.4.cz \(\chi_{504}(187, \cdot)\) n/a 568 2

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(504))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(504)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(126))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(252))\)\(^{\oplus 2}\)